Coulomb Exchange Term Research Articles

Odd-even mass differences of well and rigidly deformed nuclei in the rare earth region: A test of a newly proposed fit of average pairing matrix elements

Abstract We discuss a test of a recently proposed approach to determine average pairing matrix elements within a given interval of single-particle states (sp) around the Fermi level $\lambda$ as obtained in the so-called uniform gap method (UGM). It takes stock of the crucial role played by the averaged sp level density $\tilde{\rho}(e)$. These matrix elements are deduced within the UGM approach, from microscopically calculated $\tilde{\rho}(e)$ and gaps obtained from analytical formulae of a semi-classical nature. Two effects generally ignored in similar fits have been taken care of. They are: (a) the correction for a systematic bias in choosing to fit pairing gaps corresponding to equilibrium deformation solutions as discussed by M"{o}ller and Nix [Nucl. Phys. A 476, 1 (1992)] and (b) the correction for a systematic spurious enhancement of $\tilde{\rho}(e)$ for protons in the vicinity of $\lambda$, because of the local Slater approximation used for the treatment of the Coulomb exchange terms in most calculations (see e.g. [Phys. Rev C 84, 014310 (2011)]). This approach has been deemed to be very efficient upon performing Hartree-Fock + BCS (with seniority force and self-consistent blocking when dealing with odd nuclei) calculations of a large sample of well and rigidly deformed even-even rare-earth nuclei. The reproduction of their experimental moments of inertia has been found to be at least of the same quality as what has been obtained in a direct fit of these data [Phys. Rev C 99, 064306 (2019)]. We extend here the test of our approach to the reproduction, in the same region, of three-point odd-even mass differences centered on odd-$N$ or odd-$Z$ nuclei. The agreement with the data is again roughly of the same quality as what has been obtained in a direct fit, as performed in [Phys. Rev C 99, 064306 (2019)].

Alternative CNDOL Fockians for fast and accurate description of molecular exciton properties.

CNDOL is an a priori, approximate Fockian for molecular wave functions. In this study, we employ several modes of singly excited configuration interaction (CIS) to model molecular excitation properties by using four combinations of the one electron operator terms. Those options are compared to the experimental and theoretical data for a carefully selected set of molecules. The resulting excitons are represented by CIS wave functions that encompass all valence electrons in the system for each excited state energy. The Coulomb-exchange term associated to the calculated excitation energies is rationalized to evaluate theoretical exciton binding energies. This property is shown to be useful for discriminating the charge donation ability of molecular and supermolecular systems. Multielectronic 3D maps of exciton formal charges are showcased, demonstrating the applicability of these approximate wave functions for modeling properties of large molecules and clusters at nanoscales. This modeling proves useful in designing molecular photovoltaic devices. Our methodology holds potential applications in systematic evaluations of such systems and the development of fundamental artificial intelligence databases for predicting related properties.

The CNDOL Fockian with the configuration interaction of single excited wave functions to model the exciton properties of large molecular systems for photovoltaic devices

The approximate CNDOL Fockian, together with the configuration interaction of single excitations ([CIS|CNDOL/21]), is applied to investigate the electronic properties of three different kinds of donors (D) considered suitable for photovoltaic devices, either as isolated systems or complexed with fullerene (C60) as potential acceptors (A). Calculated charge maps, together with the Coulomb-exchange term of CIS lowest transitions, are used to understand the electronic topology and possible behaviour of D-A excitons. The effects of varying donor structures are compared. The lowest excitation energies of the donor-C60 complexes arise from electron densities localised at the central ring units in the donors. Nevertheless, C60 does not always participate in electron sharing upon excitation. It sometimes acts only as a structural companion for enhancing or not the exciton capability to provide adequate photoelectric power conversion efficiencies. The [CIS|CNDOL/21] method is validated as a valuable tool for modelling structure-related properties of molecular excitons and for predicting the performance of D–A materials in relatively big supramolecular systems. It is shown that multi-electron CIS wavefunctions are at least as useful as one-electron molecular orbitals to describe charge-transfer properties of excitons and provide a physical picture of the whole electron density.

Tensor and Coulomb-exchange terms in the relativistic mean-field model with δ -meson and isoscalar-isovector coupling

Tensor couplings and Coulomb-exchange terms in the relativistic mean-field (RMF) model impact the neutron skin thickness of nuclei [N. Liliani, A. M. Nugraha, J. P. Diningrum, and A. Sulaksono, Phys. Rev. C 93, 054322 (2016)], and a significantly thicker skin of $^{208}\mathrm{Pb}$ predicted by PREX-II leads to the existence of a phase transition in the interior of neutron stars (NSs) [F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, Phys. Rev. Lett. 120, 172702 (2018)]. In this work, we revisit the effects of tensors and Coulomb-exchange terms within the RMF model with the $\ensuremath{\delta}$-meson and isovector-isoscalar coupling on finite nuclei. We also perform a systematic investigation of the implicit effect of both terms on the nuclear matter and NS properties. We confirmed in our previous work [N. Liliani, A. M. Nugraha, J. P. Diningrum, and A. Sulaksono, Phys. Rev. C 93, 054322 (2016)] that the roles of tensor couplings, Coulomb exchange, and isovector-isoscalar coupling on the bulk properties of finite nuclei are crucial. Conversely, the $\ensuremath{\delta}$-meson term's inclusion yields no significant effect on the bulk properties of finite nuclei. The tensor couplings and Coulomb-exchange terms implicitly influence the isoscalar and isovector parts of nuclear matter. By contrast, the $\ensuremath{\delta}$-meson and isovector-isoscalar coupling significantly impact the isovector parts of nuclear matter predictions. Except for the binding energy and the equation of state of pure neutron matter in low-density cases, the nuclear matter property predictions of all parameter sets used are relatively compatible with experimental data and the chiral effective field theory result. Moreover, the tensor couplings, Coulomb-exchange, $\ensuremath{\delta}$-meson, and isovector-isoscalar coupling terms play a significant role in predicting the canonical NS radius but not in the maximum mass of NS. For relatively large ${\ensuremath{\eta}}_{2\ensuremath{\rho}}$, the compatibility of the canonical NS radius predicted by the model with all of the observation results used in this work is better than that with the tensor couplings and Coulomb exchange excluded. Moreover, the presence of tensor couplings, Coulomb exchange, and various isovector-isoscalar nonlinear coupling terms influence the relation between NS tidal deformation and neutron skin thickness of $^{208}\mathrm{Pb}$. However, tensor couplings and Coulomb-exchange terms play a crucial role in making the NS tidal deformability prediction compatible with the constraints from observations and experimental data. In conclusion, tensor couplings and Coulomb-exchange RMF models have an essential role in finite nuclei predictions and implicitly influence the nuclear matter and NS property predictions.

Systematic study of Coulomb displacement energy based on the Coulomb energy empirical formula and mass model

In this study, starting from the nuclear binding energy, we studied the empirical formula of the Coulomb energy systematically. Beginning with the simplest empirical formula, we added the Coulomb exchange term, pairing correlation, and shell correction, to calculate the Coulomb energy more precisely. In addition, we investigated the Coulomb displacement energy of widely used mass models. The discrepancies between the calculation results and experimental data might be caused by the missing of the isospin breaking terms of strong interactions in theory. By investigating these discrepancies, we found that the results obtained by microscopic theories presented a distinctive systematical behavior. Thus, these theories can be improved by including isospin breaking terms. However, in the results obtained by phenomenological models, the systematical behavior is unclear. This occurs because it is difficult for the information of the microscopic nuclear structure to be absorbed into the parameters of phenomenological models.

Self-consistent relativistic quasiparticle random-phase approximation and its applications to charge-exchange excitations

The self-consistent quasiparticle random-phase approximation (QRPA) approach is formulated in the canonical single-nucleon basis of the relativistic Hatree-Fock-Bogoliubov (RHFB) theory. This approach is applied to study the isobaric analog states (IAS) and Gamov-Teller resonances (GTR) by taking Sn isotopes as examples. It is found that self-consistent treatment of the particle-particle residual interaction is essential to concentrate the IAS in a single peak for open-shell nuclei and the Coulomb exchange term is very important to predict the IAS energies. For the GTR, the isovector pairing can increase the calculated GTR energy, while the isoscalar pairing has an important influence on the low-lying tail of the GT transition. Furthermore, the QRPA approach is employed to predict nuclear $\beta$-decay half-lives. With an isospin-dependent pairing interaction in the isoscalar channel, the RHFB+QRPA approach almost completely reproduces the experimental $\beta$-decay half-lives for nuclei up to the Sn isotopes with half-lives smaller than one second. Large discrepancies are found for the Ni, Zn, and Ge isotopes with neutron number smaller than $50$, as well as the Sn isotopes with neutron number smaller than $82$. The potential reasons for these discrepancies are discussed in detail.

Effect of exact Coulomb-exchange calculations on band-head spectra of odd-proton nuclei

Previous calculations of band-head energy spectra of odd-mass heavy nuclei in the Hartree-Fock-plus-Bardeen-Cooper-Schrieffer (HF-BCS) framework showed that the agreement with data is better for odd-neutron as compared to odd-proton nuclei. The reason for a poorer agreement with data for the latter have been ascribed to the possible usage of the Slater approximation in calculating the Coulomb-exchange term. In this work, we report the effect of exact Coulomb-exchange calculations on band-head energy spectra of two odd-proton nuclei (namely 237 Np and 241 Am) as compared to the results obtained using the Slater approximation. We performed self-consistent blocking calculations while taking the breaking of time-reversal symmetry at the mean-field level into account due to the unpaired nucleon. The SkM* and SIII parametrizations of the Skyrme interaction have been employed to approximate the effective nucleon-nucleon interaction while a seniority force is used for the pairing channel. Contrary to what was expected, our preliminary results show no improvement on the band-head spectra as compared to data when the Coulomb-exchange term is calculated exactly.

Influence of the Coulomb exchange term on nuclear single-proton resonances

Nuclear single-proton resonances are sensitive to the Coulomb field, while the exchange term of Coulomb field is usually neglected due to its nonlocality. By combining the complex scaling method with the relativistic mean-field model, the influence of the Coulomb exchange term on the single-proton resonances is investigated by taking Sn isotopes and $$N\,{=}\,82$$ isotones as examples. It is found that the Coulomb exchange term reduces the single-proton resonance energy within the range of 0.4–0.6 MeV and leads to similar isotopic and isotonic trends of the resonance energy as those without the Coulomb exchange term. Moreover, the single-proton resonance width is also reduced by the Coulomb exchange term, whose influence generally decreases with the increasing neutron number and increases with the increasing proton number. However, the influence of the Coulomb exchange term cannot change the trend of the resonance width with respect to the neutron number and proton number. Furthermore, the influence of the Coulomb exchange term on the resonance width is investigated for the doubly magic nuclei $$^{40}$$ Ca, $$^{56,78}$$ Ni, $$^{100,132}$$ Sn, and $$^{208}$$ Pb. It is found that the Coulomb exchange term reduces the proton resonance width within 0.2 MeV, whose magnitude depends on the specific nucleus and the quantum numbers of resonant states.

Impacts of the tensor couplings ofωandρmesons and Coulomb-exchange terms on superheavy nuclei and their relation to the symmetry energy

We have studied the effects of tensor coupling of $\omega$ and $\rho$ meson terms, Coulomb exchange term in local density approximation and various isoscalar-isovector coupling terms of relativistic mean field model on the properties of nuclear matter, finite nuclei, and super-heavy nuclei. We found that for the same fixed value of symmetry energy $J$ or its slope $L$ the presence of tensor coupling of $\omega$ and $\rho$ meson terms and Coulomb exchange term yields thicker neutron skin thickness of $^{208}$Pb. We also found that the roles of tensor coupling of $\omega$ and $\rho$ meson terms, Coulomb exchange term in local density approximation and various isoscalar-isovector coupling terms on the bulk properties of finite nuclei varies depending on the corresponding nucleus mass. However, on average, tensor coupling terms play a significant role in predicting the bulk properties of finite nuclei in a quite wide mass range especially in binding energies. We also observed that for some particular nuclei, the corresponding experimental data of binding energies is rather less compatible with the presence of Coulomb exchange term in local density approximation and they tend to disfavor the presence of isoscalar-isovector coupling term with too high $\Lambda$ value. Furthermore, we have found that these terms influence the detail properties of $^{292}$120 super-heavy nucleus such as binding energies, the magnitude of two nucleon gaps, single particle spectra, neutron densities, neutron skin thicknesses and mean square charge radii. However, the shell closure predictions of $^{208}$Pb and $^{292}$120 nuclei is not affected by the presence of these terms.

Splitting between bright and dark excitons in transition metal dichalcogenide monolayers

The optical properties of transition metal dichalcogenide monolayers such as the two-dimensional semiconductors MoS$_2$ and WSe$_2$ are dominated by excitons, Coulomb bound electron-hole pairs. The light emission yield depends on whether the electron-hole transitions are optically allowed (bright) or forbidden (dark). By solving the Bethe Salpeter Equation on top of $GW$ wave functions in density functional theory calculations, we determine the sign and amplitude of the splitting between bright and dark exciton states. We evaluate the influence of the spin-orbit coupling on the optical spectra and clearly demonstrate the strong impact of the intra-valley Coulomb exchange term on the dark-bright exciton fine structure splitting.

Probing single-proton resonances in nuclei by the complex-scaling method

By combining the complex scaling method with relativistic mean-field theory, single-proton resonances are probed for spherical nuclei. The energy and width are shown to decrease with the increasing neutron number for the Sn isotopes and increase with the increasing proton number for the $N=82$ isotones. Furthermore, the influence of the Coulomb exchange term is investigated for the doubly magic nuclei $^{40}\mathrm{Ca}$, $^{56,78}\mathrm{Ni}$, $^{100,132}\mathrm{Sn}$, and $^{208}\mathrm{Pb}$. It is found that the Coulomb exchange term reduces the proton resonance energy within the range of 0.4--0.6 MeV, which depends on the particular nucleus as well as the quantum numbers of the resonant state.

Coulomb and spin-orbit interactions in random-phase approximation calculations

We present a fully self-consistent computational framework composed by Hartree-Fock plus random-phase approximation where the spin-orbit and Coulomb terms of the interaction are included in both steps of the calculations. We study the effects of these terms of the interaction on the random-phase approximation calculations, where they are usually neglected. We carry out our investigation of excited states in spherical nuclei of oxygen, calcium, nickel, zirconium, tin, and lead isotope chains. We use finite-range effective nucleon-nucleon interactions of Gogny type. The size of the effects we find is, usually, of a few hundred keV. There are not simple approximations which can be used to simulate these effects since they strongly depend on all the variables related to the excited states, angular momentum, parity, excitation energy, isoscalar, and isovector characters. Even the Slater approximation developed to account for the Coulomb exchange terms in the Hartree-Fock approximation is not valid in random-phase approximation calculations.

Slater approximation for Coulomb exchange effects in nuclear covariant density functional theory

The relativistic local density approximation (LDA) for the Coulomb exchange functional in nuclear systems is presented. This approximation is composed of the well-known Slater approximation in the non-relativistic scheme and the corrections due to the relativistic effects. Its validity in finite nuclei is examined by comparing with the exact treatment of the Coulomb exchange term in the relativistic Hartree-Fock-Bogoliubov theory. The relativistic effects are found to be important and the exact Coulomb exchange energies can be reproduced by the relativistic LDA within 5% demonstrated by the semi-magic Ca, Ni, Zr, Sn, and Pb isotopes from proton drip line to neutron drip line.

Nuclear effective charge factor originating from covariant density functional theory

Guiding by the relativistic local density approximation, we explore a phenomenological formula for the coupling strength of Coulomb field to take into account the Coulomb exchange term effectively in the relativistic Hartree approximation. Its validity in finite nuclei is examined by comparing with the exact treatment of the Coulomb exchange energies in the relativistic Hartree-Fock-Bogoliubov approach. It is found that the exact Coulomb exchange energies can be reproduced by employing the phenomenological formula with the relative deviations less than 1% for semi-magic Ca, Ni, Sn, and Pb isotopes. Furthermore, we check the applicability of the phenomenological formula for the effective interactions in the relativistic Hartree approach by investigating the binding energy differences of mirror nuclei.

FURTHER MICROSCOPIC STUDIES OF THE FISSION BARRIERS OF HEAVY NUCLEI

Two systematic sources of error in most current microscopic evaluations of fission-barrier heights are studied. They are concerned with an approximate treatment of the Coulomb exchange terms (known as the Slater approximation) in the self-consistent mean-fields and the projection on good parity states (e.g., of positive parity for the spontaneous fission of an even–even nucleus) of left–right reflection asymmetric intrinsic solutions (e.g., around the second barrier). Approximate or unprojected solutions are shown to lead each to an underestimation of the barrier heights by a few hundred keV.

Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis.: (VI) hfodd (v2.40h): A new version of the program

Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis.: (VI) hfodd (v2.40h): A new version of the program

Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. VIII. Role of Coulomb exchange

Following suggestions that the energy associated with Coulomb correlations and a possible charge-symmetry breaking of nuclear forces might largely cancel the Coulomb-exchange term, we refit the HFB-14 mass model without the Coulomb-exchange term to essentially all the mass data. The resulting mass model, HFB-15, gives a better fit to the 2149 mass data, ${\ensuremath{\sigma}}_{\mathrm{rms}}$ falling from 0.729 to 0.678 MeV. The improvement in the energy differences between mirror nuclei is particularly striking: the Nolen-Schiffer anomaly, which is strong for HFB-14, is essentially eliminated. As for the extrapolation to highly neutron-rich nuclei, the HFB-15 model differs significantly from HFB-14, with up to 15 MeV less binding being predicted. However, the differences in the predicted values of differential quantities such as the neutron-separation energies, \ensuremath{\beta}-decay energies and fission barriers are very much smaller.

Identification of theg92proton and neutron band crossing in theN=ZnucleusSr76

High-spin states in $^{76}\mathrm{Sr}$ have been studied using Gammasphere plus Microball detector arrays. The known yrast band has been extended beyond the first band crossing, which involves the simultaneous alignment of pairs of ${\mathrm{g}}_{\frac{9}{2}}$ protons and neutrons, to a tentative spin of $24\ensuremath{\hbar}$. The data are compared with the results of cranked relativistic mean-field (CRMF) and cranked relativistic Hartree-Bogoliubov (CRHB) calculations. The properties of the band, including the ${\mathrm{g}}_{\frac{9}{2}}$ proton/neutron band crossing frequency and moments of inertia, are found to be well reproduced by the CRHB calculations. Furthermore, the unpaired CRMF calculations show quite good agreement with the data beyond the band crossing region, indicating that pairing is weak at these frequencies. The high spin results suggest that there is little evidence for an isoscalar $(t=0)$ $\mathit{np}$ pair field. Moreover, a systematic study of the band crossings in even-even $N=Z$ nuclei for the first time reveals that there is no evidence to support the existence of the Coulomb antipairing effect caused by the Coulomb exchange term.

Displacement energies with the Skyrme Hartree–Fock method

The binding energy differences of mirror nuclei are calculated with the Skyrme Hartree–Fock method and compared to experiment. The Nolen–Schiffer anomaly can be accounted for either by dropping the Coulomb exchange term or by introducing a charge-symmetry breaking interaction. The displacement energy systematics can be quantitatively reproduced with either of these modifications. The results are important for an accurate description of the proton and neutron drip lines. The possible origins of these modifications in terms of nuclear correlations and nucleon-nucleon charge-symmetry breaking interactions are discussed.

Exchange effects in the transition amplitude for inelastic electron-cluster collisions

The effect of the electronic exchange in the calculation of the scattering amplitude for inelastic electron-cluster collisions is evaluated. The target cluster initially in the ground state is excited, by means of the Coulomb forces, into a particle-hole state. An antisymmetrized transition amplitude is evaluated in the scope of the distorted-wave Born approximation. A comparison is made with a calculation in which the exchange is taken into account by adding to the direct interaction a Coulomb exchange term in the Slater approximation. The treatments are applied to the closed-shell Na{sub 8} cluster for incident energies up to 5 eV. {copyright} {ital 1997} {ital The American Physical Society}